Excursion Laplace Exponents Under Height Truncation
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis manuscript studied one-dimensional diffusions reflected at a boundary and analyze their pathwise “episodes” away from the boundary through Itô’s excursion theory.
The manuscript does more than derive formulas. It explains how the excursion Poisson point process makes extreme value questions essentially 1D once the excursion tail intensity is known, and it connects extremes and cumulative accumulation through a single Levy measure. All in all, it makes a great contribution to the study on the Excursion Laplace Exponents.
Author Response
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Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript entitled “Excursion Laplace Exponents under Height Truncation” presents a solid mathematical contribution by providing closed-form formulas for the Laplace exponents of height-truncated Itô excursions. The use of additive functionals and the treatment of reflected one-dimensional diffusions are rigorous. However, to enhance the impact and accessibility of the work for the applied mathematics community, I recommend the following revisions:
- Context and Practical Relevance
Although theoretical rigor is the focus, the author would benefit from expanding the introduction and conclusion with application examples. Specifically, the usefulness of the functional and the truncation should be discussed from a risk management perspective. In financial barrier models or physical systems with tolerance limits, height truncation represents the exclusion of catastrophic events. A discussion on how these results alter risk perception compared to non-truncated models would add substantial value.
- Implementation and Numerical Inversion
Obtaining Laplace transforms is an important milestone, but the practical application of these formulas depends on their inversion to the time domain. The author is requested to comment on the numerical stability of the derived expressions. Is it feasible to use algorithms such as Abate–Whitt or Stehfest to recover probability densities from the proposed functions?
- Illustration with Canonical Processes
It is recommended to include a numerical case study using a classical diffusion, such as the Ornstein–Uhlenbeck (OU) process or the Feller process. Graphically demonstrating how the Laplace exponent behaves as a function of the truncation height and the decay rate would allow the reader to visualize the model’s sensitivity.
- Notation Structure and Clarity
Given the dense nature of the proofs, introducing a summary of notation or a glossary of variables (especially distinguishing Itô measures from probability measures ) would improve readability. The transition between the Lemmas and the main Theorem could be smoother if brief explanatory paragraphs were added to convey the probabilistic intuition behind each intermediate step.
- Minor Errors and Formal Rigor
- Check the consistency of the indices in the integrals of the additive functionals throughout Sections 2 and 3.
- Ensure that the boundary conditions for the "boundary flux" are explicitly defined for the case of diffusions with singularities at the origin.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper "Excursion Laplace Exponents under Height Truncation" by Tristan Guillaume investigates one-dimensional reflected diffusions using Itô's excursion theory. It focuses on characterizing the duration and cumulative load of "episodes" (excursions) that stay below a specific height threshold a. This is a novel approach in several key aspects:
- He introduce a new analytic object: the height-truncated Itô-excursion Laplace exponent
- The paper demonstrates high analytical proficiency by deriving explicit, closed-form solutions for three major solvable families of diffusions
- Deriving explicit laws for maxima and order statistics of episode loads.
The numerical simulation also supports these finding. I propose an acceptance for publication if some following minor revisions can be adressed:
- consistent italicized a
- diagrammatic representation: a sample path of a reflected diffusion, highlighting an excursion that stays below a versus one that crosses a
- numerical validation sections: it would be helpful to include some plot of excursions and a small table or a clearly itemized list of the parameters used (e.g., the specific values for mu, theta and sigma. This ensures that other researchers can exactly reproduce the results.
Author Response
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Author Response File: Author Response.pdf
